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From Enderton's book Elements of set theory.

Example 2, first chapter $\phi$ is the empty subset:

$\lbrace \phi\rbrace\in\lbrace\lbrace\phi\rbrace\rbrace$, but $\lbrace \phi\rbrace\nsubseteq\lbrace\lbrace\phi\rbrace\rbrace$.

Why the second statement is not true? We know that $\phi\subseteq A$, where $A$ is any subset, but why $\phi\in A$ is false?

What is the intuitive answer?

Asaf Karagila
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user2820579
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    ${{\emptyset}}$ is a set containing exactly one element, and that element is ${\emptyset}$ just the same as ${a}$ is a set containing exactly one element, namely $a$. In the same way that ${a}$ does not contain the element $b$ (where $a$ is considered different than $b$) the set ${{\emptyset}}$ does not contain $\emptyset$ as an element (despite having it as a subset). Having something as an element is different than having something as a subset. – JMoravitz Apr 18 '17 at 23:46
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    I assume the $A$ that you ask about is ${{\phi}}$. This set has exactly one member, namely ${\phi}$, because this is the only thing named between the outermost braces. The one member of $A$, ${\phi}$, is not $\phi$, and therefore $\phi$ is not a member of $A$. (If you wonder why ${\phi}$ is not $\phi$, notice that ${\phi}$ has a member, namely $\phi$, whereas $\phi$ has no members.) – Andreas Blass Apr 18 '17 at 23:47
  • Perhaps part of the frustration is that you might be thinking of looking as many layers down as you like, but we are only looking one layer down. The set ${{a,b},c}$ for example contains exactly two elements, the first element is ${a,b}$ and the second element is $c$, however ${{a,b},c}$ does not contain the element $a$ (despite it appearing several layers down). – JMoravitz Apr 18 '17 at 23:49
  • subsets aren't elements. {{emptyset}} has one element. It is {emptyset} which is not emptyset. {emptyset} has one element. It is emptyset. So {emptyset} has an element that {{emptyset}} does not. Consider emptyset = a. And consider {a} = b. Then ${a} \not \subset {b}$ because {a} has an elemet that {b} does not. – fleablood Apr 18 '17 at 23:57
  • It's abstract but sets are collections of things and are not the things themselves. My house is a collection of many things. Two of those things are my two cats; fleabottom and gunkbert. But {fleabottom, gunkber} is a collection. That is not something that is in my house. But {fleabottom, gunkbert} is a subcollect of my things. So is the emptyset. But nowhere in my houst do I actually have the emptyset. – fleablood Apr 19 '17 at 00:05

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